HairBSDF, gamma for a refracted ray

?

Percentage Accurate: 57.7% → 97.0%
Time: 54.5s
Precision: binary32
Cost: 6688

?

\[\left(\left(-1 \leq sinTheta_O \land sinTheta_O \leq 1\right) \land \left(-1 \leq h \land h \leq 1\right)\right) \land \left(0 \leq eta \land eta \leq 10\right)\]
\[\sin^{-1} \left(\frac{h}{\sqrt{eta \cdot eta - \frac{sinTheta_O \cdot sinTheta_O}{\sqrt{1 - sinTheta_O \cdot sinTheta_O}}}}\right) \]
↓
\[\sin^{-1} \left(\frac{h}{\mathsf{fma}\left(-0.5, \frac{sinTheta_O}{\frac{eta}{sinTheta_O}}, eta\right)}\right) \]
(FPCore (sinTheta_O h eta)
 :precision binary32
 (asin
  (/
   h
   (sqrt
    (-
     (* eta eta)
     (/
      (* sinTheta_O sinTheta_O)
      (sqrt (- 1.0 (* sinTheta_O sinTheta_O)))))))))
↓
(FPCore (sinTheta_O h eta)
 :precision binary32
 (asin (/ h (fma -0.5 (/ sinTheta_O (/ eta sinTheta_O)) eta))))
float code(float sinTheta_O, float h, float eta) {
	return asinf((h / sqrtf(((eta * eta) - ((sinTheta_O * sinTheta_O) / sqrtf((1.0f - (sinTheta_O * sinTheta_O))))))));
}
↓
float code(float sinTheta_O, float h, float eta) {
	return asinf((h / fmaf(-0.5f, (sinTheta_O / (eta / sinTheta_O)), eta)));
}

function code(sinTheta_O, h, eta)
	return asin(Float32(h / sqrt(Float32(Float32(eta * eta) - Float32(Float32(sinTheta_O * sinTheta_O) / sqrt(Float32(Float32(1.0) - Float32(sinTheta_O * sinTheta_O))))))))
end
↓
function code(sinTheta_O, h, eta)
	return asin(Float32(h / fma(Float32(-0.5), Float32(sinTheta_O / Float32(eta / sinTheta_O)), eta)))
end

\sin^{-1} \left(\frac{h}{\sqrt{eta \cdot eta - \frac{sinTheta_O \cdot sinTheta_O}{\sqrt{1 - sinTheta_O \cdot sinTheta_O}}}}\right)
↓
\sin^{-1} \left(\frac{h}{\mathsf{fma}\left(-0.5, \frac{sinTheta_O}{\frac{eta}{sinTheta_O}}, eta\right)}\right)

Local Percentage Accuracy vs ?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Herbie found 3 alternatives:

AlternativeAccuracySpeedup

Accuracy vs Speed

The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Bogosity?

Bogosity

Derivation?

  1. Initial program 57.4%

    \[\sin^{-1} \left(\frac{h}{\sqrt{eta \cdot eta - \frac{sinTheta_O \cdot sinTheta_O}{\sqrt{1 - sinTheta_O \cdot sinTheta_O}}}}\right) \]

  2. Taylor expanded in sinTheta_O around 0 94.8%

    \[\leadsto \sin^{-1} \left(\frac{h}{\color{blue}{eta + -0.5 \cdot \frac{{sinTheta_O}^{2}}{eta}}}\right) \]

  3. Simplified96.9%

    \[\leadsto \sin^{-1} \left(\frac{h}{\color{blue}{\mathsf{fma}\left(-0.5, \frac{sinTheta_O}{\frac{eta}{sinTheta_O}}, eta\right)}}\right) \]
    Step-by-step derivation

    [Start]94.8%

    \[ \sin^{-1} \left(\frac{h}{eta + -0.5 \cdot \frac{{sinTheta_O}^{2}}{eta}}\right) \]

    +-commutative [=>]94.8%

    \[ \sin^{-1} \left(\frac{h}{\color{blue}{-0.5 \cdot \frac{{sinTheta_O}^{2}}{eta} + eta}}\right) \]

    fma-def [=>]94.8%

    \[ \sin^{-1} \left(\frac{h}{\color{blue}{\mathsf{fma}\left(-0.5, \frac{{sinTheta_O}^{2}}{eta}, eta\right)}}\right) \]

    unpow2 [=>]94.8%

    \[ \sin^{-1} \left(\frac{h}{\mathsf{fma}\left(-0.5, \frac{\color{blue}{sinTheta_O \cdot sinTheta_O}}{eta}, eta\right)}\right) \]

    associate-/l* [=>]96.9%

    \[ \sin^{-1} \left(\frac{h}{\mathsf{fma}\left(-0.5, \color{blue}{\frac{sinTheta_O}{\frac{eta}{sinTheta_O}}}, eta\right)}\right) \]

  4. Final simplification96.9%

    \[\leadsto \sin^{-1} \left(\frac{h}{\mathsf{fma}\left(-0.5, \frac{sinTheta_O}{\frac{eta}{sinTheta_O}}, eta\right)}\right) \]

Alternatives

Alternative 1
Accuracy97.0%
Cost6688
\[\sin^{-1} \left(\frac{h}{\mathsf{fma}\left(-0.5, \frac{sinTheta_O}{\frac{eta}{sinTheta_O}}, eta\right)}\right) \]
Alternative 2
Accuracy94.5%
Cost3552
\[\sin^{-1} \left(\frac{h}{eta + -0.5 \cdot \frac{sinTheta_O \cdot sinTheta_O}{eta}}\right) \]
Alternative 3
Accuracy93.5%
Cost3296
\[\sin^{-1} \left(\frac{h}{eta}\right) \]

Reproduce?

herbie shell --seed 2023256 
(FPCore (sinTheta_O h eta)
  :name "HairBSDF, gamma for a refracted ray"
  :precision binary32
  :pre (and (and (and (<= -1.0 sinTheta_O) (<= sinTheta_O 1.0)) (and (<= -1.0 h) (<= h 1.0))) (and (<= 0.0 eta) (<= eta 10.0)))
  (asin (/ h (sqrt (- (* eta eta) (/ (* sinTheta_O sinTheta_O) (sqrt (- 1.0 (* sinTheta_O sinTheta_O)))))))))